Some Aspects of Hankel Matrices in Coding Theory and Combinatorics

Ulrich Tamm Department of Computer Science University of Chemnitz 09107 Chemnitz, Germany [email protected]

Submitted: December 8, 2000; Accepted: May 26, 2001. MR Subject Classiﬁcations: primary 05A15, secondary 15A15, 94B35

Abstract Hankel matrices consisting of Catalan numbers have been analyzed by various authors. Desainte- Q i+j+2n Catherine and Viennot found their determinant to be 1≤i≤j≤k i+j and related them to the Q i+j−1+2n Bender - Knuth conjecture. The similar determinant formula 1≤i≤j≤k i+j−1 can be shown 2m+1 to hold for Hankel matrices whose entries are successive middle binomial coefficients m . Generalizing the Catalan numbers in a different direction, it can be shown that determinants of 1 3m+1 Hankel matrices consisting of numbers 3m+1 m yield an alternate expression of two Mills – Robbins – Rumsey determinants important in the enumeration of plane partitions and alternating sign matrices. Hankel matrices with determinant 1 were studied by Aigner in the definition of Catalan – like numbers. The well - known relation of Hankel matrices to orthogonal polynomials further yields a combinatorial application of the famous Berlekamp – Massey algorithm in Coding Theory, which can be applied in order to calculate the coefficients in the three – term recurrence of the family of orthogonal polynomials related to the sequence of Hankel matrices.

I. Introduction A Hankel matrix (or persymmetric matrix)   c0 c1 c2 ... cn−1    c1 c2 c3 ... cn   c c c ... c  An =  2 3 4 n+1  . (1.1)  . . . .   . . . .  cn−1 cn cn+1 ... c2n−2 is a matrix (aij) in which for every r the entries on the diagonal i + j = r are the same, i.e., ai,r−i = cr for some cr. the electronic journal of combinatorics 8 2001, #A1 1 For a sequence c0,c1,c2,... of real numbers we also consider the collection of Hankel (k) matrices An , k =0, 1,..., n =1, 2,...,where   ck ck+1 ck+2 ... ck+n−1    ck+1 ck+2 ck+3 ... ck+n  (k)  c c c ... c  An =  k+2 k+3 k+4 k+n+1  . (1.2)  . . . .   . . . .  ck+n−1 ck+n ck+n+1 ... ck+2n−2 So the parameter n denotes the size of the matrix and the 2n − 1 successive elements ck,ck+1,...,ck+2n−2 occur in the diagonals of the Hankel matrix. We shall further denote the determinant of a Hankel matrix (1.2) by

(k) (k) dn =det(An ). (1.3) Hankel matrices have important applications, for instance, in the theory of moments, and in Padé approximation. In Coding Theory, they occur in the Berlekamp - Massey algorithm for the decoding of BCH - codes. Their connection to orthogonal polynomials often yields useful applications in Combinatorics: as shown by Viennot [76] Hankel determinants enumerate certain families of weighted paths, Catalan – like numbers as defined by Aigner [2] via Hankel determinants often yield sequences important in combinatorial enumeration, and as a recent application, they turned out to be an important tool in the proof of the refined alternating sign matrix conjecture. The framework for studying combinatorial applications of Hankel matrices and further aspects of orthogonal polynomials was set up by Viennot [76]. Of special interest are 1 2m+1 determinants of Hankel matrices consisting of Catalan numbers 2m+1 m .Desainte– (k) Catherine and Viennot [24] provided a formula for det(An )andalln ≥ 1, k ≥ 0incase c that the entries m are Catalan numbers, namely c 1 2m+1 m , ,... For the sequence m = 2m+1 m , =0 1 of Catalan numbers it is Y i j n d(0) d(1) ,d(k) + +2 k ≥ ,n≥ . . n = n =1 n = i j for 2 1 (1 4) 1≤i≤j≤k−1 + Desainte–Catherine and Viennot [24] also gave a combinatorial interpretation of this determinant in terms of special disjoint lattice paths and applications to the enumeration of Young tableaux, matchings, etc.. Q i+j−1+c c They studied (1.4) as a companion formula for 1≤i≤j≤k i+j−1 , which for integer was shown by Gordon (cf. [67]) to be the generating function for certain Young tableaux. For even c =2n this latter formula also can be expressed as a Hankel determinant formed 2m+1 of successive binomial coefficients m . c 2m+1 m , ,... For the binomial coefficients m = m , =0 1 Y i j − n d(0) ,d(k) + 1+2 k, n ≥ . . n =1 n = i j − for 1 (1 5) 1≤i≤j≤k + 1 the electronic journal of combinatorics 8 2001, #A1 2 We are going to derive the identities (1.4) and (1.5) simultaneously in the next section. Our main interest, however, concerns a further generalization of the Catalan numbers and their combinatorial interpretations. In Section III we shall study Hankel matrices whose entries are defined as generalized c 1 3m+1 Catalan numbers m = 3m+1 m . In this case we could show that nY−1 j j j Yn 6j−2 d(0) (3 + 1)(6 )!(2 )!,d(1) 2j . . n = j j n = 4j−1 (1 6) j=0 (4 + 1)!(4 )! j=1 2 2j These numbers are of special interest, since they coincide with two Mills – Robbins – Rum- sey determinants, which occur in the enumeration of cyclically symmetric plane partitions and alternating sign matrices which are invariant under a reflection about a vertical axis. The relation between Hankel matrices and alternating sign matrices will be discussed in Section IV. Let us recall some properties of Hankel matrices. Of special importance is the equation       c0 c1 c2 ... cn−1 an,0 −cn        c1 c2 c3 ... cn   an,1   −cn+1         c2 c3 c4 ... cn+1  ·  an,2  =  −cn+2  . (1.7)  . . . .   .   .   . . . .   .   .  cn−1 cn cn+1 ... c2n−2 an,n−1 −c2n−1

(0) It is known (cf. [16], p. 246) that, if the matrices An are nonsingular for all n, then the polynomials

j j−1 j−2 tj(x):=x + aj,j−1x + aj,j−2x + ...aj,1x + aj,0 (1.8) form a sequence of monic orthogonal polynomials with respect to the linear operator T l l mapping x to its moment T (x )=cl for all l,i.e.

T (tj(x) · tm(x)) = 0 for j =6 m. (1.9) and that

m T (x · tj(x))=0form =0,...,j− 1. (1.10)

In Section V we shall study matrices Ln =(l(m, j))m,j=0,1,...,n−1 deﬁned by

m l(m, j)=T (x · tj(x)) (1.11) By (1.10) these matrices are lower triangular. The recursion for Catalan – like numbers, as defined by Aigner [2] yielding another generalization of Catalan numbers, can be derived via matrices Ln with determinant 1. Further, the Lanczos algorithm as discussed in [13] t yields a factorization Ln = An · Un,whereAn is a nonsingular Hankel matrix as in (1.1), Ln is defined by (1.11) and the electronic journal of combinatorics 8 2001, #A1 3   100... 00    a1,0 10... 00  a a ...  Un =  2,0 2,1 1 00 . (1.12)  . . . . .   . . . . .  an−1,0 an−1,1 an−2,2 ... an−1,n−2 1 is the triangular matrix whose entries are the coefficients of the polynomials tj(x), j = 0,...,n− 1. In Section V we further shall discuss the Berlekamp – Massey algorithm for the decoding of BCH–codes, where Hankel matrices of syndromes resulting after the transmission of a code word over a noisy channel have to be studied. Via the matrix Ln defined by (1.11) it will be shown that the Berlekamp – Massey algorithm applied to Hankel matrices with real entries can be used to compute the coefficients in the corresponding orthogonal polynomials and the three – term recurrence defining these polynomials.

Several methods to ﬁnd Hankel determinants are presented in [61]. We shall mainly concentrate on their occurrence in the theory of continued fractions and orthogonal polynomials. If not mentioned otherwise, we shall always assume that all Hankel matrices An under consideration are nonsingular. Hankel matrices come into play when the power series

2 F (x)=c0 + c1x + c2x + ... (1.13) (0) (1) is expressed as a continued fraction. If the Hankel determinants dn and dn are diﬀerent from 0 for all n the so–called S–fraction expansion of 1 − xF (x) has the form

c0x 1 − xF (x)=1− q x . (1.14) − 1 1 e1x 1 − q x − 2 1 e2x 1 − 1 − ... (k) Namely, then (cf. [55], p. 304 or [78], p. 200) for n ≥ 1 and with the convention d0 =1 for all k it is

d(1) · d(0) d(0) · d(1) q n n−1 ,e n+1 n−1 . . n = (1) (0) n = (0) (1) (1 15) dn−1 · dn dn · dn For the notion of S– and J– fraction (S stands for Stieltjes, J for Jacobi) we refer to the standard books by Perron [55] and Wall [78]. We follow here mainly the (qn,en)–notation of Rutishauser [65]. 1 For many purposes it is more convenient to consider the variable x in (1.13) and study power series of the form

the electronic journal of combinatorics 8 2001, #A1 4 c c c 1 F 1 0 1 2 ... . x (x)= x + x2 + x3 + (1 16) and its continued S–fraction expansion

c0 q1 x − e − 1 1 q2 x − e − 2 1 x − ... which can be transformed to the J–fraction

c0 (1.17) β1 x − α1 − β2 x − α2 − β3 x − α3 − x − α4 − ... with α1 = q1,andαj+1 = qj+1 + ej,βj = qjej for j ≥ 1. (cf. [55], p.375 or [65], pp. 13). The J–fraction corresponding to (1.14) was used by Flajolet ([26] and [27]) to study combinatorial aspects of continued fractions, especially he gave an interpretation of the coefficients in the continued fractions expansion in terms of weighted lattice paths. This interpretation extends to parameters of the corresponding orthogonal polynomials as studied by Viennot [76]. For further combinatorial aspects of orthogonal polynomials see e.g. [28], [72]. Hankel determinants occur in Padé approximation and the determination of the eigenvalues of a matrix using their Schwarz constants, cf. [65]. Especially, they have been studied by Stieltjes in the theory of moments ([70], [71]). He stated the problem to find out if a measure µ exists such that

Z ∞ l x dµ(x)=cl for all l =0, 1,... (1.18) 0 R dµ(t) P∞ c ,c ,c ,... − l cl . for a given sequence 0 1 2 by the approach x+t = l=0( 1) xl+1 Stieltjes could show that such a measure exists if the determinants of the Hankel ma- (0) (1) trices An and An are positive for all n. Indeed, then (1.9) results from the quality of pj (x) tj x the approximation to (1.16) by quotients of polynomials tj (x) where ( ) are just the polynomials (1.8). Hence they obey the three – term recurrence

tj(x)=(x − αj)tj−1(x) − βj−1 · tj−2(x),t0(x)=1,t1(x)=x − α1, (1.19) where

α1 = q1, and αj+1 = qj+1 + ej,βj = qjej for j ≥ 1. (1.20) the electronic journal of combinatorics 8 2001, #A1 5 In case that we consider Hankel matrices of the form (1.2) and hence the corresponding 2 power series ck + ck+1x + ck+2x + ..., we introduce a superscript (k) to the parameters in question. (k) (k) Hence, qn and en denote the coeﬃcients in the continued fractions expansions c c k , k q(k)x e(k)q(k) − 1 x − q(k) − 1 1 1 (k) 1 (k) (k) e x (k) (k) e q 1 − 1 x − q − e − 2 2 q(k)x 2 1 x − q(k) − e(k) − ... 1 − 2 3 2 1 − ... and

(k) j (k) j−1 (k) j−2 (k) (k) tj (x)=x + aj,j−1x + aj,j−2x + ...aj,1 x + aj,0 are the corresponding polynomials obeying the three – term recurrence

(k) (k) (k) (k) (k) tj (x)=(x − αj )tj−1(x) − βj−1tj−2(x). Several algorithms are known to determine this recursion. We mentioned already the Berlekamp – Massey algorithm and the Lanczos algorithm. In the quotient–diﬀerence (k) (k) algorithm due to Rutishauser [65] the parameters qn and en are obtained via the so– called rhombic rule

(k) (k) (k+1) (k) (k) en = en−1 + qn − qn ,e0 = 0 for all k, (1.21)

e(k+1) c q(k) q(k+1) · n ,q(k) k+1 k. . n+1 = n (k) 1 = for all (1 22) en ck

II. Hankel Matrices and Chebyshev Polynomials Let us illustrate the methods introduced by computing determinants of Hankel matrices whose entries are successive Catalan numbers. In several recent papers (e.g. [2], [47], [54], [62]) these determinants have been studied under various aspects and formulae were given for specialQ parameters. Desainte–Catherine and Viennot in [24] provided the general d(k) i+j+2n n k solution n = 1≤i≤j≤k−1 i+j for all and . This was derived as a companion formula (yielding a “90 % bijective proof” for tableaux whose columns consist of an even number of elements and are bounded by height 2n) to Gordon’s result [36] in the proof of the Q c+i+j−1 Bender – Knuth conjecture [8]. Gordon proved that 1≤i≤j≤k i+j−1 is the generating function for Young tableaux with entries from {1,...,n} strictly increasing in rows and not decreasing in columns consisting of ≤ c columns and largest part ≤ k. Actually, this follows from the more general formula in the Bender – Knuth conjecture by letting q → 1, see also [67], p. 265. By refining the methods of [24], Choi and Gouyou – Beauchamps [21] could also derive Gordon’s formula for even c =2n. In the following proposition we shall apply a well - the electronic journal of combinatorics 8 2001, #A1 6 known recursion for Hankel determinants allowing to see that in this case also Gordon’s formula can be expressed as a Hankel determinant, namely the matrices then consist of 2m+1 consecutive binomial coefficients of the form m . Simultaneously, this yields another proof of the result of Desainte – Catherine and Viennot, which was originally obtained by application of the quotient – difference algorithm [77]. Proposition 2.1: c 1 2m+1 m , ,... a) For the sequence m = 2m+1 m , =0 1 of Catalan numbers it is Y i j n d(0) d(1) ,d(k) + +2 k ≥ ,n≥ . . n = n =1 n = i j for 2 1 (2 1) 1≤i≤j≤k−1 + c 2m+1 m , ,... b) For the binomial coefficients m = m , =0 1 Y i j − n d(0) ,d(k) + 1+2 k, n ≥ . . n =1 n = i j − for 1 (2 2) 1≤i≤j≤k + 1 Proof: The proof is based on the following identity for Hankel determinants.

(k+1) (k−1) (k+1) (k−1) (k) 2 dn · dn − dn−1 · dn+1 − [dn ] =0. (2.3) This identity can for instance be found in the book by Polya and Szegö [59], Ex. 19, p. 102. It is also an immediate consequence of Dodgson’s algorithm for the evaluation of determinants (e.g. [82]). We shall derive both results simultaneously. The proof will proceed by induction on n+k. (k) It is well known, e.g. [69], that for the Hankel matrices An with Catalan numbers as (0) (1) entries it is dn = dn = 1. For the induction beginning it must also be verified that d(2) n d(3) (n+1)(n+2)(2n+3) n = +1andthat n = 6 is the sum of squares, cf. [47], which can also be easily seen by application of recursion (2.3). A(k) 2k+1 2k+3 Furthermore, for the matrix n whose entries are the binomial coefficients k , k+1 , (0) (1) ... it was shown in [2] that dn =1anddn =2n + 1. Application of (2.3) shows that d(2) (n+1)(2n+1)(2n+3) n = 3 , i. e., the sum of squares of the odd positive integers. c Also, it is easily seen by comparing successive quotients k+1 that for n = 1 the product in ck (2.1) yields the Catalan numbers and the product in (2.2) yields the binomial coefficients 2k+1 k+1 , cf. also [24]. Now it remains to be verified that (2.1) and (2.2) hold for all n and k, which will be done by checking recursion (2.3). The sum in (2.3) is of the form (with either d = 0 for (2.1) or d = 1 for (2.2) and shifting k to k + 1 in (2.1))

Yk i j − d n kY−2 i j − d n Yk i j − d n kY−2 i j − d n − + +2 · + +2 − + +2( +1)· + +2( 1)− i j − d i j − d i j − d i j − d i,j=1 + i,j=1 + i,j=1 + i,j=1 +

 2 kY−1 i j − d n −  + +2  i j − d i,j=1 + the electronic journal of combinatorics 8 2001, #A1 7  2 kY−1 i j − d n  + +2  · = i j − d i,j=1 + ! Qk Qk−1 Qk−1 Qk−1 (k + j − d +2n) · (k − 1+j − d) (j − d +2n) · (k − 1+j − d) j=1 j=1 − j=0 j=1 − Qk Qk−1 Qk Qk−1 1 j=1(k + j − d) · j=1 (k − 1+j − d +2n) j=1(k + j − d) · j=1 (1 + j − d +2n)  2 kY−1 i j − d n  + +2  · = i j − d i,j=1 + (2n +2k − d)(2n +2k − 1 − d)(k − d) (2n − d)(2n +1− d)(k − d) · − − 1 . (2n + k − d)(2k − d)(2k − 1 − d) (2n + k − d)(2k − d)(2k − 1 − d) This expression is 0 exactly if

(2n +2k − d)(2n +2k − 1 − d)(k − d) − (2n − d)(2n +1− d)(k − d)− −(2n + k − d)(2k − d)(2k − 1 − d)=0. In order to show (2.1), now observe that here d = 0 and then it is easily veriﬁed that

(n + k)(2n +2k − 1) − n(2n +1)− (2n + k)(2k − 1) = 0. In order to show (2.2), we have to set d = 1 and again the analysis simpliﬁes to verifying

(2n +2k − 1)(n + k − 1) − (2n − 1)n − (2n + k − 1)(2k − 1) = 0. £ Remarks: 1) As pointed out in the introduction, Desainte–Catherine and Viennot [24] derived iden- (0) tity (2.1) and recursion (2.3) simultaneously proves (2.2). The identity det(An )=1, c 2m+1 when the m’s are Catalan numbers or binomial coefficients m can already be found (1) (2) (3) in [52], pp. 435 – 436. dn ,dn ,anddn for this case were already mentioned in the proof (4) d(3) dn n+1 of Theorem 2.1. The next determinant in this series is obtained via (4) = (3) .Forthe dn−1 dn−1 d(3) ·d(3) 2 d(4) n+1 n n(n+1) (n+2)(2n+1)(2n+3) . Catalan numbers then n = 5 = 180 2) Formula (2.1) was also studied by Desainte–Catherine and Viennot [24] in the analysis of disjoint paths in a bounded area of the integer lattice and perfect matchings in a (k) certain graph as a special Pfaffian. An interpretation of the determinant dn in (2.1) as the number of k–tuples of disjoint positive lattice paths (see the next section) was used to construct bijections to further combinatorial configurations. Applications of (2.1) in Physics have been discussed by Guttmann, Owczarek, and Viennot [40]. 3) The central argument in the proof of Theorem 2.1 was the application of recursion (2.3). Let us demonstrate the use of this recursion with another example. Aigner [3] could show that the Bell numbers are the unique sequence (cm)m=0,1,2,... such that

the electronic journal of combinatorics 8 2001, #A1 8 Yn Yn (0) (1) (2) det(An )=det(An )= k!, det(An )=rn+1 k!, (2.4) k=0 k=0 Pn where rn =1+ l=1 n(n − 1) ···(n − l + 1) is the total number of permutations of n (0) (1) things (for det(An ) and det(An ) see [27] and [23]). In [3] an approach via generating (2) (2) (2) Qn functions was used in order to derive dn =det(An ) in (2.4). Setting dn = rn+1 · k=0 k! in (2.4), with (2.3) one obtains the recurrence rn+1 =(n +1)· rn +1,r2 =5, which just characterizes the total number of permutations of n things, cf. [63], p. 16, and hence can (2) (0) (1) derive det(An )fromdet(An ) and det(An )alsothisway. Q i+j−d+2n 4) From the proof of Proposition 2.1 it is also clear that 1≤i,j≤k i+j−d yields a sequence (k) of Hankel determinants dn only for d =0, 1, since otherwise recursion (2.3) is not fulfilled. As pointed out, in [24] formula (2.1) was derived by application of the quotient – difference (k) (k) algorithm, cf. also [21] for a more general result. The parameters qn and en also can be obtained from Proposition 2.1. (k) (k) Corollary 2.1: For theP Catalan numbers the coefficients qn and en in the continued ∞ 1 2(k+m)+1 xm fractions expansion of m=0 2(k+m)+1 k+m as in (1.14) are given as (2n +2k − 1)(2n +2k) (2n)(2n +1) q(k) = ,e(k) = . (2.5) n (2n + k − 1)(2n + k) n (2n + k)(2n + k +1) 2m+1 PFor the binomial coefficients m the corresponding coefficients in the expansion of ∞ 2(k+m)+1 xm m=0 k+m are (2n +2k)(2n +2k +1) (2n − 1)(2n) q(k) = ,e(k) = . (2.6) n (2n + k − 1)(2n + k) n (2n + k)(2n + k +1) Proof: (2.5) and (2.6) can be derived by application of the rhombic rule (1.21) and (1.22). They are also immediate from the previous Proposition 2.1 by application of (1.15), which (k) for k>0 generalizes to the following formulae from [65], p. 15, where the dn ’s are Hankel determinants as (1.3).

d(k+1)d(k) d(k) d(k) q(k) n n−1 ,e(k) n+1 n−1 . n = (k) (k+1) n = (k) (k+1)

dn dn−1 dn dn £ (k) Corollary 2.2: The orthogonal polynomials associated to the Hankel matrices An of c 1 2m+1 Catalan numbers m = 2m+1 m are

(k) (k) (k) (k) (k) 4k +2 t(k)(x)=(x − α(k))t − β t (x),t(x)=1,t(x)=x − n n n−1 n−1 n−2 0 1 k +2 where

the electronic journal of combinatorics 8 2001, #A1 9 (k) 2k(k − 1) (2n +2k − 1)(2n +2k)(2n)(2n +1) α =2− ,β(k) = . n+1 (2n + k + 2)(2n + k) n (2n + k − 1)(2n + k)2(2n + k +1)

(k) (k) (k) Proof: By (1.20), βn = qn · en as in the previous corollary and

(k) (k) (2n +2k + 1)(2n +2k + 2)((2n + k)+(2n)(2n + 1)(2n + k +2) α = q + e(k) = n+1 n+1 n (2n + k + 1)(2n + k + 2)(2n + k) 8n2 +8nk +8n +2k +4k2 2k(k − 1) = =2− .

(2n + k + 2)(2n + k) (2n + k + 2)(2n + k) £ Especially for small parameters k the following families of orthogonal polynomials arise here.

(0) (0) (0) (0) (0) tn (x)=(x − 2) · tn−1(x) − tn−2(x),t0 (x)=1,t1 (x)=x − 1, (1) (1) (1) (1) (1) tn (x)=(x − 2) · tn−1(x) − tn−2(x),t0 (x)=1,t1 (x)=x − 2, 2 2 2 (n +1) + n (2) n − 1 (2) (2) (2) 5 t(2)(x)= x − · t (x) − t (x),t(x)=1,t (x)=x − . n n(n +1) n−1 n2 n−2 0 1 2 It is well - known that the Chebyshev – polynomials of the second kind

b n c X2 n − i u x − i x n−2i n( )= ( 1) i (2 ) i=0 with recursion

un(x)=2x · un−1(x) − un−2(x),u0(x)=1,u1(x)=2x come in for Hankel matrices with Catalan numbers as entries. For instance, in this case the ﬁrst orthogonal polynomials in Corollary 2.2 are

(0) 2 1 x (1) 2 1 x t (x )= u2n( ),t(x )= u2n+1( ). n x 2 n x 2 (k) Corollary 2.3: The orthogonal polynomials associated to the Hankel matrices An of c 2m+1 binomial coeﬃcients m = m are

(k) (k) (k) (k) (k) 4k +6 t(k)(x)=(x − α(k))t − β t (x),t(x)=1,t(x)=x − n n n−1 n−1 n−2 0 1 k +2 where

(k) 2k(k +1) (k) (2n +2k)(2n +2k + 1)(2n − 1)(2n) α =2− ,β= . n+1 (2n + k + 2)(2n + k) n+1 (2n + k − 1)(2n + k)2(2n + k +1) the electronic journal of combinatorics 8 2001, #A1 10 (k) (k) (k) Proof: Again, βn = qn · en as in the previous corollary and

(k) (k) (2n +2k + 2)(2n +2k + 3)((2n + k)+(2n − 1)(2n)(2n + k +2) α = q + e(k) = n+1 n+1 n (2n + k)(2n + k + 1)(2n + k +2)

8n2 +8nk +8n +2k2 +4k 2k(k +1) = =2− . (2n + k + 2)(2n + k) (2n + k + 2)(2n + k)

III. Generalized Catalan Numbers And Hankel Determinants p ≥ 1 pm+1 For an integer 2 we shall denote the numbers pm+1 m as generalized Catalan numbers. The Catalan numbers occur for p = 2. (The notion “generalized Catalan numbers” as in [42] is not standard, for instance, in [39], pp. 344 – 350 it is suggested to denote them “Fuss numbers”). Their generating function X∞ pm C x 1 +1 xm . p( )= pm m (3 1) m=0 +1 fulﬁlls the functional equation

p Cp(x)=1+x · Cp(x) , from which immediately follows that

1 p−1 =1− x · Cp(x) . (3.2) Cp(x) Further, it is X∞ pm p − C x p−1 1 + 1 xm. . p( ) = pm p − m (3 3) m=0 + 1 +1 1 pm+1

It is well known that the generalized Catalan numbers pm+1 m count the number of

× Z i, j i, j paths in the integer lattice Z (with directed vertices from ( )toeither( +1)or to (i +1,j)) from the origin (0, 0) to (m, (p − 1)m) which never go above the diagonal

p − x y × Z ( 1) = . Equivalently, they count the number of paths in Z starting in the origin (0, 0) and then ﬁrst touching the boundary {(l +1, (p − 1)l +1):l =0, 1, 2,...} in (m, (p − 1)m + 1) (cf. e.g. [75]). Viennot [76] gave a combinatorial interpretation of Hankel determinants in terms of disjoint Dyck paths. In case that the entries of the Hankel matrix are consecutive Catalan numbers this just yields an equivalent enumeration problem analyzed by Mays and Woj- ciechowski [47]. The method of proof from [47] extends to Hankel matrices consisting of generalized Catalan numbers as will be seen in the following proposition. c c 1 pm+1 Proposition 3.1: If the m’s in (1.2) are generalized Catalan numbers, m = pm+1 m , (k) p ≥ 2 a positive integer, then det(An )isthenumberofn–tuples (γ0,...,γn−1) of vertex the electronic journal of combinatorics 8 2001, #A1 11

× Z i, j – disjoint paths in the integer lattice Z (with directed vertices from ( )toeither (i, j +1)orto(i +1,j)) never crossing the diagonal (p − 1)x = y, where the path γr is from (−r, −(p − 1)r)to(k + r, (p − 1)(k + r)). Proof: The proof follows the same lines as the one in [32], which was carried out only for the case p = 2 and is based on a result in [46] on disjoint path systems in directed graphs. We follow here the presentation in [47]. Namely, let G be an acyclic directed graph and let A = {a0,...,an−1}, B = {b0,...,bn−1} be two sets of vertices in G of the same size n. A disjoint path system in (G, A, B)isa system of vertex disjoint paths (γ0,...,γn−1), where for every i =0,...,n− 1 the path γi leads from ai to bσ(i) for some permutation σ on {0,...,n− 1}. + Now let pij denote the number of paths leading from ai to bj in G,letp be the number of disjoint path systems for which σ is an even permutation and let p− be the number of disjoint path systems for which σ is an odd permutation. Then det((pij)i,j=0,...,n−1)= p+ − p− (Theorem 3 in [47]). Now consider the special graph G0 with vertex set

V { u, v ∈ × Z p − u ≤ v}, = ( ) Z :( 1) i. e. the part of the integer lattice on and above the diagonal (p − 1)x = y, and directed edges connecting (u, v)to(u, v +1)andto(u +1,v) (if this is in V,ofcourse). Further let A = {a0,...an−1} and B = {b0,...bn−1} be two sets disjoint to each other and to V. Then we connect A and B to G0 by introducing directed edges as follows

ai → (−i, −(p − 1)i), (k + i, (p − 1)(k + i)) → bi,i=0,...,n− 1. (3.4) Now denote by G00 the graph with vertex set V∪A∪Bwhose edges are those from G0 and the additional edges connecting A and B to G0 as described in (3.4). Observe that any permutation σ on {0,...,n− 1} besides the identity would yield some j and l with σ(j) >jand σ(l)

described in Proposition 3.1. £ Remarks: 1) The use of determinants in the enumeration of disjoint path systems is well known, e.g. [31]. In a similar way as in Proposition 3.1 we can derive an analogous result for the number of tuples of vertex – disjoint lattice paths, with the diﬀerence that the paths now are not allowed to touch the diagonal (p−1)x = y before they terminate in (m, (p−1)m). , m, p− m 1 pm+p−1 Since the number of such paths from (0 0) to ( ( 1) )is pm+p−1 m+1 (cf. e.g. the (k) appendix), this yields a combinatorial interpretation of Hankel matrices An with these numbers as entries as in (1.2). 2) For the Catalan numbers, i. e. p = 2, lattice paths are studied which never cross the diagonal x = y. Viennot provided a combinatorial interpretation of orthogonal polynomials

the electronic journal of combinatorics 8 2001, #A1 12 by assigning weights to the steps in such a path, which are obtained from the coefficients in the three–term recurrence of the orthogonal polynomials ([76], cf also. [26]). In the case that all coefficients αj are 0, a Dyck path arises with vertical steps having all weight 1 and horizontal steps having weight βj for some j. For the Catalan numbers as entries in the Hankel matrix all βj’s are 1, since the Chebyshev polynomials of second kind arise. So the total number of all such paths is counted. Observe that Proposition 3.1 extends the path model for the Catalan numbers in another direction, namely the weights of the single steps are still all 1, but the paths now are not allowed to cross a different boundary. In order to evaluate the Hankel determinants we further need the following identity. Lemma 3.1:Letp ≥ 2 be an integer. Then ! ! X∞ pm X∞ pm X∞ pm xm · 1 +1 xm +1 xm. . m pm m = m (3 5) m=0 m=0 +1 m=0 Proof: We are obviously done if we could show that for all m =0, 1, 2,... pm Xm pl p m − l +1 1 +1 · ( ) . m = pl l m − l l=0 +1 pm+1 In order to do so, we count the number m of lattice paths (where possible steps are from (i, j)toeither(i, j +1)orto(i +1,j)) from (0, 0) to (m, (p − 1)m + 1) in a second way. Namely each such path must go through at least one of the points (l, (p − 1)l +1), l =0, 1,...,m. Now we divide the path into two subpaths, the first subpath leading from the origin (0, 0) to the first point of the form (l, (p − 1)l + 1) and the second subpath l, p − l m, p − m 1 pl+1 from ( ( 1) +1)to( ( 1) + 1). Recall that there are pl+1 l possible choices p(m−l) for the first subpath and obviously there exist m−l possibilities for the choice of the

second subpath. £ m , , ... c 1 3m+1 b 1 3m+2 Theorem 3.1:For =01 2 let denote m = 3m+1 m and m = 3m+2 m+1 . Then   c0 c1 c2 ... cn−1    c1 c2 c3 ... cn  n−1   Y (3j + 1)(6j)!(2j)!  c2 c3 c4 ... cn+1  = ,  . . . .  (4j + 1)!(4j)!  . . . .  j=0 cn−1 cn cn+1 ... c2n−2   c1 c2 c3 ... cn    c2 c3 c4 ... cn+1  Yn 6j−2   2j  c3 c4 c5 ... cn+2  = (3.6)  . . . .  2 4j−1  . . . .  j=1 2j cn cn+1 cn+2 ... c2n−1

the electronic journal of combinatorics 8 2001, #A1 13 and   b0 b1 b2 ... bn−1    b1 b2 b3 ... bn  Yn 6j−2   2j  b2 b3 b4 ... bn+1  = ,  . . . .  2 4j−1  . . . .  j=1 2j bn−1 bn bn+1 ... b2n−2   b1 b2 b3 ... bn    b2 b3 b4 ... bn+1  n   Y (3j + 1)(6j)!(2j)!  b3 b4 b5 ... bn+2  = . (3.7)  . . . .  (4j + 1)!(4j)!  . . . .  j=0 bn bn+1 bn+2 ... b2n−1 Proof: Observe that Q Q Q Q Q m j m−1 j m−1 j m−1 2 j m−1 1 j 3m j=1(3 ) j=0 (3 +1) j=0 (3 +2) 27 j=0 ( 3 + ) j=0 ( 3 + ) = Q Q =( )m Q m m m j m−1 j m m−1 1 j ! j=1(2 ) j=0 (2 +1) 4 ! j=0 ( 2 + ) and accordingly Q Q Q Q Q m j m−1 j m−1 j m−1 2 j m−1 4 j 3m +1 j=1(3 ) j=0 (3 +4) j=0 (3 +2) 27 j=0 ( 3 + ) j=0 ( 3 + ) = Q Q =( )m Q . m m m j m−1 j m m−1 3 j ! j=1(2 ) j=0 (2 +3) 4 ! j=0 ( 2 + )

Then with (3.2) and (3.5) we have the representation P ∞ 3m xm 2 m=0 m F (α, β, γ, y) D(x):=1− x · C3(x) = P = , ∞ 3m+1 xm F α, β ,γ ,y m=0 m ( +1 +1 ) which is the quotient of two hypergeometric series, where αβ α(α +1)β(β +1) α(α +1)(α +2)β(β +1)(β +2) F (α, β, γ, y)=1+ y + y2 + y2 + ... γ 2! · γ(γ +1) 3! · γ(γ +1)(γ +2) with the parameter choice 2 1 1 27 α = ,β= ,γ= ,y= x. (3.8) 3 3 2 4 For quotients of such hypergeometric series the continued fractions expansion as in (1.14) was found by Gauss (see [55], p. 311 or [78], p. 337). Namely for n =1, 2,... it is (α + n)(γ − β + n) (β + n)(γ − α + n) en = ,qn = . (γ +2n)(γ +2n +1) (γ +2n − 1)(γ +2n)

(D) (D) Now denoting by qn and en the coefficients in the continued fractions expansion of 2 the power series D(x)=1− xC3(x) under consideration, then taking into account that y 27 x = 4 we obtain with the parameters in (3.8) that the electronic journal of combinatorics 8 2001, #A1 14 3 (6n + 1)(3n +2) 3 (6n − 1)(3n +1) e(D) = ,q(D) = . (3.9) n 2 (4n + 1)(4n +3) n 2 (4n − 1)(4n +1) 2 2 The continued fractions expansion of 1 + xC3(x) differs from that of 1 − xC3(x) only by changing the sign of c0 in (1.14). (0) (1) So, by application of (1.15) the identity (3.7) for the determinants dn and dn of Hankel 1 3m+2 matrices with the numbers 3m+2 m+1 as entries is easily verified by induction. Namely, observe that 3 (6n − 1)(3n +1) 2(6n)(6n − 1)(2n)(3n +1) = 2 (4n − 1)(4n +1) (4n + 1)(4n)2(4n − 1) 4n−1 (1) (0) n n n 2 dn d (3 + 1)(6 )!(2 )! · 2n · n−1 = n n 6n−2 = (1) (0) (4 + 1)!(4 )! 2n dn−1 dn and that 3 (6n + 1)(3n +2) (6n + 4)(6n + 3)(6n + 2)(6n + 1)(2n +1) = 2 (4n + 1)(4n +3) 2(4n + 3)(4n +2)2(4n + 1)(3n +1) 6n+4 (0) (1) (4n + 1)!(4n)! d d = 2n+2 · = n+1 · n−1 , 4n+3 n n n (0) (1) 2 2n+1 (3 + 1)(6 )!(2 )! dn dn (0) (1) (0) (1) (0) where dn−1,dn−1,dn ,dn ,dn+1 are the determinants for the Hankel matrices in (3.7). In order to find the determinants for the Hankel matrices in (3.6) with generalized Catalan 1 3m+1 2 1 D x − xC3 x numbers 3m+1 m as entries, just recall that ( )=1 ( ) = C3(x) .Sothe continued fractions expansion of −x −x xC x − − 1+ 3( )=1 2 =1 (C) 1 − xC3(x) q x 1 − 1 e(C)x 1 − 1 q(C)x 1 − 2 1 − ... q(C) (C) (D) (C) (D) is obtained by setting 1 =1,en = qn for n ≥ 1andqn = en−1 for n ≥ 2. £ (k) Problem: In the last section we were able to derive all Hankel determinants dn with p Catalan numbers as entries. So the case = 2 for Hankel determinants (1.2) consisting of 1 pm+1 p d(0) numbers pm+1 m is completely settled. For = 3, the above theorem yields n and (1) (k) dn . However the methods do not work in order to determine dn for k ≥ 2. Also they do not allow to find determinants of Hankel matrices consisting of generalized Catalan numbers when p ≥ 4. What can be said about these cases? Let us finally discuss the connection to the Mills – Robbins – Rumsey determinants

the electronic journal of combinatorics 8 2001, #A1 15 ! 2Xn−2 i + µ j 2j−t Tn(x, µ)=det x , (3.10) t − i 2j − t t=0 i,j=0,...,n−1 where µ is a nonnegative integer (discussed e.g. in [50], [6], [5], [22], and [57]). For µ =0, 1 (µ) it is Tn(1,µ)=dn - the Hankel determinants in (3.6). This coincidence does not continue for µ ≥ 2. Using former results by Andrews [4], Mills, Robbins, and Rumsey [50] could derive that µ i j nY−1 T ,µ + + 1 µ . n(1 )=det j − i = n ∆2k(2 )(311) 2 i,j=0,...,n−1 2 k=0 where ∆0(µ)=2andwith(x)j = x(x +1)(x +2)···(x + j − 1) µ k 1 µ k 3 ( +2 +2)k( 2 +2 + 2 )k−1 ∆2k(µ)= ,k>0. k 1 µ k 3 ( )k( 2 + + 2 )k−1 They also state that the proof of formula (3.11) is quite complicated and that it would be interesting to find a simpler one. One might look for an approach via continued fractions for further parameters µ, however, application of Gauss’s theorem only works for µ =0, 1, where (3.9) also follows from (3.11). Mills, Robbins, and Rumsey [50] found the number of cyclically symmetric plane partitions of size n, which are equal to its transpose–complement to be the determinant Tn(1, 0). They also conjectured Tn(x, 1) to be the generating function for alternating sign matrices invariant under a reflection about a vertical axis, especially Tn(1, 1) should then be the total number of such alternating sign matrices as stated by Stanley [68]. We shall further discuss this conjecture in SectionP IV. T ,µ 2n−2 i+µ j , The determinant n(1 )=det t=0 t−i t−j comes in as counting func- i,j=0,...,n−1 tion for another class of vertex–disjoint path families in the integer lattice. Namely, for such a such a tuple (γ0,...,γn−1) of disjoint paths, path γi leads from (i, 2i + µ)to(2i, i). By a bijection to such disjoint path families for µ = 0 the enumeration problem for the above – mentioned family of plane partitions was finally settled in [50]. IV. Alternating Sign Matrices An alternating sign matrix is a square matrix with entries from {0, 1, −1} such that i) the entries in each row and column sum up to 1, ii) the nonzero entries in each row and column alternate in sign. An example is   000 1000    100−1001    000 1000    010−1010 (4.1)    000 1000  001−1100 000 1000 the electronic journal of combinatorics 8 2001, #A1 16 Robbins and Rumsey discovered the alternating sign matrices in the analysis of Dodgson’s algorithm in order to evaluate the determinant of an n × n – matrix. Reverend Charles Lutwidge Dodgson, who worked as a mathematician at the Christ College at the University of Oxford is much wider known as Lewis Carroll, the author of [18]. His algorithm, which is presented in [16], pp. 113 – 115, is based on the following identity for any matrix ([25], for a combinatorial proof see [82]).

det ((ai,j)i,j=1,...,n) · det ((ai,j)i,j=2,...,n−1)=det((ai,j)i,j=1,...,n−1) · det ((ai,j)i,j=2,...,n) −

−det ((ai,j)i=1,...,n−1,j=2,...,n) · det ((ai,j)i=2,...,n,j=1,...,n−1) . (4.2)

If (ai,j)i,j=1,...,n in (4.2) is a Hankel matrix, then all the other matrices in (4.2) are Hankel matrices, too. Hence recursion (2.3) from the introduction is an immediate consequence of Dodgson’s result. In the course of Dodgson’s algorithm only 2 × 2 determinants have to be calculated. Robbins asked what would happen, if in the algorithm we would replace the determinant evaluation aijai+1,j+1 − ai,j+1ai+1,j by the prescription aijai+1,j+1 + xai,j+1ai+1,j,wherex is some variable. a It turned out that this yields a sum of monomials in the ij and their inverses,Q each n bij monomial multiplied by a polynomial in x. The monomials are of the form i,j=1 aij where the bij ’s are the entries in an alternating sign matrix. The exact formula can be found in Theorem 3.13 in the book “Proofs and Confirmations: The Story of The Alternating Sign Matrix Conjecture” by David Bressoud [16]. The alternating sign matrix conjecture concerns the total number of n × n alternating Qn−1 (3j+1)! sign matrices, which was conjectured by Mills, Robbins, and Rumsey to be j=0 (n+j)! . The problem was open for fifteen years until it was finally settled by Zeilberger [80]. The development of ideas is described in the book by Bressoud. There are deep relations to various parts of Algebraic Combinatorics, especially to plane partitions, where the same counting function occurred, and also to Statistical Mechanics, where the configuration of water molecules in “square ice” can be described by an alternating sign matrix. As an important step in the derivation of the refined alternating sign matrix conjecture c 1−qm+1 [81], a Hankel matrix comes in, whose entries are m = 1−q3(m+1) . The relevant orthogonal polynomials in this case are a discrete version of the Legendre polynomials. Many problems concerning the enumeration of special types of alternating sign matrices are still unsolved, cf. [16], pp. 201. Some of these problems have been presented by Stanley in [68], where it is also conjectured that the number V (2n + 1) of alternating sign matrices of odd order 2n + 1 invariant under a reflection about a vertical axis is Yn 6j−2 2j V (2n +1)= 4j−1 j=1 2 2j

A more refined conjecture is presented by Mills, Robbins, and Rumsey [50] relating this type of alternating sign matrices to the determinant Tn(x, 1) in (3.10). Especially, the electronic journal of combinatorics 8 2001, #A1 17 Q 6j−2 T , n ( 2j ) V n n(1 1) = j=1 2 4j−1 is conjectured to be the total number (2 +1). Aswesawin ( 2j ) (1) Section III, the same formula comes in as the special Hankel determinant dn ,wherein 1 3m+1 (1.2) we choose generalized Catalan numbers 3m+1 m as entries. Let us consider this conjecture a little closer. If an alternating sign matrix (short: ASM) is invariant under a reflection about a vertical axis, it must obviously be of odd order 2n + 1, since otherwise there would be a row containing two successive nonzero entries with the same sign. For the same reason, such a matrix cannot contain any 0 in its central column as seen in the example (4.1). In [15], cf. also [16], Ch. 7.1, an equivalent counting problem via a bijection to families of disjoint paths in a square lattice is presented. Denote the vertices corresponding to the entry aij in the ASM by (i, j), i, j =0,...,n− 1. Then following the outermost path from (n − 1, 0) to (0,n− 1), the outermost path in the remaining graph from (0,n− 2) to (n − 2, 0), and so on until the path from (0, 1) to (1, 0) one obtains a collection of lattice paths, which are edge-disjoint but may share vertices. Since there can be no entry 0 in the central column of the ASM invariant under a reflection about a vertical axis, the entries a0,n,a2,n,a4,n,...,a2n,n must be 1 and a1,n = a3,n = a5,n = ...a2n,n = −1. This means that for i =0,...n− 1 the path from (2n − i, 0) to (0, 2n − i) must go through (2n − i, n) where it changes direction from East to North and after that in (2n − i − 1,n) it again changes direction to East and continues in (2n − i − 1,n+1). Because of the reflection–invariance about the central column the matrix of size (2n + 1) × (2n + 1) is determined by its column numbers. n +1,n +2,...2n.So,bythe above considerations the matrix can be reconstructed from the collection of subpaths (µ0,µ1,...,µn−1)whereµi leads from (2n − i − 1,n+1)to(0, 2n − i). By a reflection about the horizontal and a 90 degree turn to the left, we now map the

ν × Z collection of these paths to a collection of paths ( 0,ν1,...,νn−1) the integer lattice Z , such that the inner most subpath in the collection leads from (−1, 0) to (0, 0) and path νi leads from (−2i − 1, 0) to (0,i). Denoting by vi,s the y–coordinate of the s-th vertical step (where the path is followed from the right to the left) in path number i, i =1,...,n− 1–pathν0 does not contain vertical steps – the collection of paths (ν0,ν1,...,νn−1) can be represented by a two–dimensional array (plane partition) of positive integers

vn−1,1 vn−1,2 vn−1,2 ... vn−1,n−2 vn−1,n−1 vn−2,1 vn−2,2 ... vn−2,n−2 . . . . (4.3) v2,1 v2,2 v1,1

with weakly decreasing rows, i. e. vi,1 ≥ vi,2 ≥ ... ≥ vi,i for all i, and the following restrictions:

1) 2i − 1 ≤ vi,1 ≤ 2i + 1 for all i =1,...,n− 1,

2) vi,s − vi,s−1 ≤ 1 for all i, s with s>i.

the electronic journal of combinatorics 8 2001, #A1 18 3) vi+1,i+1 ≥ vi,i for all 1 ≤ i ≤ n − 1. So for n = 1 there is only the empty array and for n = 2 there are the three possibilities v1,1 =1,v1,1 =2,orv1,1 =3.Forn = 3 the following 26 arrays obeying the above restrictions exist:

31 32 33 41 42 43 44 51 52 53 54 1 1 1 1 1 1 1 1 1 1 1

55 42 43 44 52 53 54 55 53 54 55 1 2 2 2 2 2 2 2 3 3 3 32 33 43 44 2 2 3 3

Now consider a collection (γ0,γ1,...,γn−1) of vertex disjoint paths in the integer lattice as required in Theorem 3.1, where the single paths are not allowed to cross the diagonal 2x = y and path γi leads from (−i, −2i)to(i+1, 2i+2). Obviously, the initial segment of path γi must be the line connecting (−i, −2i)and(−i, i+2). Since no variation is possible in this part, we can remove these initial segments and obtain a collection (η0,...,ηn−1) of vertex–disjoint paths, where now ηi leads from (−i, i +2)to(i +1, 2i +2).

We now denote by vi,s the position of the s-th vertical step (i. e. the number of the horizontal step before the s–th vertical step in the path counted from right to left) in path ηi, i =1,...,n− 1 and obtain as a representation of the collection (η0,...,ηn−1)a two–dimensional array of positive integers with weakly decreasing rows as in (4.3), where the restrictions now are:

1) 2i − 1 ≤ vi,1 ≤ 2i + 1 for all i =1,...,n,

2’) vi,s − vi,s−1 ≤ 2 for all i, s with s>i. Again, for n = 1 there is only the empty array and for n = 2 there are the three choices v1,1 =1,v1,1 =2,orv1,1 = 3 as above. For n = 3 the ﬁrst 22 arrays above also fulﬁll the conditions 2’), whereas the four arrays in the last row do not. However, they can be replaced by

41 51 51 52 2 2 3 3 in order to obtain a total number of 26 as above. Unfortunately, we did not ﬁnd a bijection between these two types of arrays or the corresponding collections of paths yet.

V. Catalan – like Numbers and the Berlekamp – Massey Algorithm In this section we shall study two – dimensional arrays l(m, j), m, j =0, 1, 2,... and the matrices Ln =(l(m, j))m,j=0,1,...,n−1 deﬁned by

m l(m, j)=T (x · tj(x)), (5.1)

the electronic journal of combinatorics 8 2001, #A1 19 where T is the linear operator deﬁned under (1.9). Application of the three–term– recurrence (1.19)

tj(x)=(x − αj)tj−1(x) − βj−1tj−2(x)

and the linearity of T gives the recursion

l(m, j)=l(m − 1,j+1)+αj+1l(m − 1,j)+βjl(m − 1,j− 1) (5.2)

with initial values l(m, 0) = cm, l(0,j) = 0 for j =0(and6 β0 =0,ofcourse). Especially, cf. also [78], p. 195,

l(m, m)=c0β1β2 ···βm,l(m +1,m)=c0β1β2 ···βm(α1 + α2 + ...+ αm+1)(5.3)

We shall point out two connections of the matrices Ln to Combinatorics and Coding Theory. Namely, for the case that βj = 1 for all j the matrices Ln occur in the derivation of Catalan – like numbers as defined by Aigner in [2]. They also can be determined in t order to find the factorization Ln = An·Un,whereAn is a nonsingular Hankel matrix of the form (1.1) and Un is the matrix (1.12) with the coefficients of the orthogonal polynomials in (1.8). Via formula (5.3) the Berlekamp – Massey algorithm can be applied to find the parameters αj and βj in the three – term recurrence of the orthogonal polynomials (1.8). Aigner in [2] introduced Catalan – like numbers and considered Hankel determinants consisting of these numbers. For positive reals a, s1,s2,s3,... Catalan – like numbers (a,~s) Cm , ~s =(s1,s2,s3,...) can be defined as entries b(m, 0) in a two – dimensional array b(m, j), m =0, 1, 2,..., j =0, 1,...,m, with initial conditions b(m, m) = 1 for all m = 0, 1, 2,..., b(0,j) = 0 for j>0, and recursion

b(m, 0) = a · b(m − 1, 0) + b(m − 1, 1),

b(m, j)=b(m − 1,j− 1) + sj · b(m − 1,j)+b(m − 1,j+ 1) for j =1,...,m. (5.4)

The matrices Bn =(b(m, j))m,j=0,...,n−1, obtained from this array, have the property that t Bn · Bn is a Hankel matrix, which has, of course, determinant 1, see also [66] for the Catalan numbers.

The matrices Bn can be generalized in several ways. For instance, with βj = 1 for all j ≥ 2, α1 = a and αj+1 = sj for j ≥ 2 the recursion (5.2) now yields the matrix Ln =(l(m, j)m,j=0,...,n−1). Another generalization of the matrices Bn will be mentioned below. s s j Aigner [2] was especially interested in Catalan – like numbers with j = for all and s C(a,s) 2m+1 some ﬁxed denoted here by m . In the example below the binomial coeﬃcients m (3,2) arise as Cm .

the electronic journal of combinatorics 8 2001, #A1 20 1 31 10 5 1 35 21 7 1 126 84 36 9 1 (a,~s) So, by the previous considerations, choosing cm = Cm we have that the determinant (0) (1) dn = 1 for all n. In [2] it is also computed the determinant dn via the recurrence

(1) (1) (1) dn = sn−1 · dn−1 − dn−2. (1) (1) with initial values d0 =1,d1 = a.

Remarks:

1) One might introduce a new leading element c−1 to the sequence c0,c1,c2,...and deﬁne (−1) (−1) the n × n Hankel matrix An and its determinant dn for this new sequence. Let (s,s) (cm = Cm )m=0,1,... be the sequence of Catalan–like numbers with parameters (s, s), (k) s>1andletc−1 =1.LetAn be the Hankel matrix of size n × n as under (1.2) and let (k) dn denote its determinant. Then

Xn+1 (−1) (0) (1) (2) 2 dn =(s − 1)(n − 1) + 1,dn =1,dn = sn +1,dn = (sj +1) . j=1

(0) (1) This result follows, since dn and dn are known from Propositions 6 and 7 in [2]. So the (k) (−1) (2) sequences dn are known for two successive k’s, such that the formulae for dn and dn are easily found using recursion (2.3). (1,1) (2,2) 2) In [2] it is shown that Cm are the Motzkin numbers, Cm are the Catalan numbers (3,3) and Cm are restricted hexagonal numbers. Guy [41] gave an interpretation of the (4,4) numbers Cm starting with 1, 4, 17, 76, 354,.... They come into play when determining the number of walks in the three – dimensional integer lattice from (0, 0, 0) to (i, j, k) k i, j terminating at height , which never√ go below the ( )–plane. With the results of [2] 1−4x− 1−8x+12x2 their generating function is 2x2 .

Lower triangular matrices Ln as deﬁned by (5.1) are also closely related to the Lanczos algorithm. Observe that with (5.3) we obtain the parameters in the three – term recursion in a form which was already known to Chebyshev in his algorithm in [19], p. 482, namely

l(1, 0) l(j +1,j) l(j, j − 1) l(j, j) α1 = and αj+1 = − ,βj = for j ≥ 1. l(0, 0) l(j, j) l(j − 1,j− 1) l(j − 1,j− 1) (5.5)

the electronic journal of combinatorics 8 2001, #A1 21 Since further l(m, 0) = cm for all m ≥ 0 by (5.3) it is l(m − 1, 1) = l(m, 0) − α1l(m − 1, 0) and

l(m − 1,j+1)=l(m, j) − αj+1l(m − 1,j) − βjl(m − 1,j− 1) for j>0, from which the following recursive algorithm is immediate.   00... 00 c0      10... 00  c1    ~l   Z  01... 00 n − × Starting with 1 =  .  and deﬁning =   of size (2 1) .  . . . .  c . . . . 2n−2 00... 10 (2n − 1) and Zt its transpose, we obtain recursively

~ t ~ ~ ~ t ~ ~ ~ l1 = Z · l0 − α1l0, lj+1 = Z · lj − αj+1 · lj − βj · lj−1 for j>0 ~ The subvectors of the initial n elements of lj+1 then form the (j + 1)–th column (j = 1,...,n− 2) of Ln. t In a similar way the matrix Un, the transpose of the matrix (1.12) consisting  of the 1    0  coeﬃcients of the orthogonal polynomials, can be constructed. Here ~u0 =  .  is the  .  0 ﬁrst unit column vector of size 2n − 1 and then the further columns are obtained via

~u1 = Z · ~u0 − α1 · ~u0,~uj+1 = Z · ~uj − αj+1 · u~j − βj · ~uj−1 t Again the ﬁrst n elements of ~uj form the j–th column of Un. t This is the asymmetric Lanczos algorithm yielding the factorization An · Un = Ln as studied by Boley, Lee, and Luk [13], where An is an n × n Hankelmatrixasin(1.1). Their work is based on a former paper by Phillips [58]. The algorithm is O(n2) due to the t fact that the columns in Ln and Un are obtained only using the entries in the previous two columns. t The symmetric Lanczos algorithm in [13] yields the factorization An = Mn · Dn · Mn. −1 Here, cf. [13], p. 120, Ln = Mn · Dn where Mn = Un is the inverse of Un and Dn is the diagonal matrix with the eigenvalues of An. A combinatorial interpretation of the matrix Mn was given by Viennot [76].

When Dn is the identity matrix, then Ln = Mn and the matrix Mn was used in [54] to derive combinatorial identities as for Catalan – like numbers. Namely, in [54], the −1 Stieltjes matrix Sn = Mn · M n was applied, where M n =(mn+1,j)m,j=0,...,n−1 for Mn = (mn,j)m,j=0,...,n−1.Then

the electronic journal of combinatorics 8 2001, #A1 22   α0 100... 0    β0 α1 10... 0   β α ...  Sn =  0 1 2 1 0   . . . . .   . . . . .  0000... αn−1 is tridiagonal with the parameters of the three – term recurrence on the diagonals. Important for the decoding of BCH codes, studied in the following, is also a decomposition t of the Hankel matrix An = VnDnVn as a product of a Vandermonde matrix Vn,its t transpose Vn and the diagonal matrix Dn. Here the parameters in the Vandermonde matrix are essentially the roots of the polynomial tn(x). This decomposition was known already to Baron Gaspard Riche de Prony [60] (rather known as the leading engineer in the construction of the Pont de la Concorde in Paris and as project head of the group producing the logarithmic and trigonometric tables from 1792 - 1801), cf. also [14].

Let us now discuss the relation of the Berlekamp – Massey algorithm to orthogonal polynomials. Via (5.3) the parameters rj in the Berlekamp – Massey algorithm presented below will be explained in terms of the three – term recurrence of the orthogonal polynomials related to An. Peterson [56] and Gorenstein and Zierler [38] presented an algorithm for the decoding of BCH codes. The most time–consuming task is the inversion of a Hankel matrix An as in (1.1), in which the entries ci now are syndromes resulting after the transmission of a codeword over a noisy channel. Matrix inversion, which takes O(n3) steps was proposed to solve equation (1.7). 2 Berlekamp found a way to determine the an,j in (1.7) in O(n ) steps. His approach was to determine them as coeﬃcients of a polynomial u(x) which is found as appropriate solution of the “key equation”

F (x)u(x)=q(x)modx2t+1.

Here the coefficients c0,...,c2t up to degree 2t of F (x) can be calculated from the received word. Further, the roots of u(x) yield the locations of the errors (and also determine q(x)). By the application in Coding Theory one is interested in finding polynomials of minimum possible degree fulfilling the key equation. This key equation is solved by iteratively k+1 calculating solutions (qk(x),uk(x)) to F (x)uk(x)=qk(x)modz , k =0,...,2t. Massey [48] gave a variation of Berlekamp’s algorithm in terms of a linear feedback shift register. The algorithm is presented by Berlekamp in [9]. We follow here Blahut’s book [11], p. 180. The algorithm consist in constructing a sequence of shift registers (`j,uj(x)), j =1,... , 2n − 2, where `j denotes the length (the degree of uj)and

j j−1 uj(x)=bj,jx + bj,j−1x + ...+ bj,1x +1. the electronic journal of combinatorics 8 2001, #A1 23 the feedback–connection polynomial of the j–th shift register. For an introduction to shift registers see, e.g., [11], pp. 131, The Berlekamp – Massey algorithm works over any ﬁeld and will iteratively compute the polynomials uj(x) as follows using a second sequence of polynomials vj(x).

Berlekamp – Massey Algorithm (as in [11], p. 180): Let u0(x)=1,v0(x)=1and `0 = 0. Then for j =1,...,2n − 2set

X`j rj = bj−1,tcj−1−t, (5.6) t=0

`j = δj(j − `j−1)+(1− δj)`j−1, (5.7) uj(x) 1 −rjx uj−1(x) = · , (5.8) vj(x) δj · 1/rj (1 − δj)x vj−1(x) where 1ifrj =0and26 `j−1 ≤ j − 1 δj = . (5.9) 0 otherwise Goppa [33] introduced a more general class of codes (containing the BCH – codes as special case) for which decoding is based on the solution of the key equation F (x)u(x)=q(x) mod G(x) for some polynomial G(x). Berlekamp’s iterative algorithm does not work for arbitrary polynomial G(x) (cf. [10]). Sugiyama et al. [73] suggested to solve this new key equation by application of the Euclidean algorithm for the determination of the greatest common divisor of F (x)andG(x), where the algorithm stops, when the polynomials u(x) and q(x) of appropriate degree are found. They also showed that for BCH codes the Berlekamp algorithm usually has a better performance than the Euclidean algorithm. A decoding procedure based on continued fractions for separable Goppa codes was presented by Goppa in [34] and later for general Goppa codes in [35]. The relation of Berlekamp’s algorithm to continued fraction techniques was pointed out by Mills [49] and thoroughly studied by Welch and Scholtz [79].

Cheng [20] analysed that the sequence `j provides the information when Berlekamp’s algorithm completes one iterative step of the continued fraction, which happens when `

rj j−m uj(x)=uj−1(x) − x um−1(x). rm

the electronic journal of combinatorics 8 2001, #A1 24 Here rm and rj are different from 0 and rm+1 = ... = rj−1 = 0, which means that in (5.7) δm+1 = ... = δj−1 =0andδj = 1, such that at time j for the first time after m anew shift register must be designed. This fact can be proved inductively as in [12], p. 374. An approach reflecting the mathematical background of these “jumps” via the Iohvidov index of the Hankel matrix or the block structure of the Padé table is carried out by Jonckheere and Ma [44]. Several authors (e.g. [45], p. 156, [43], [44], [13]) point out that the proof of the above recurrence is quite complicated or that there is need for a transparent explanation. We shall see now that the analysis is much simpler for the case that all principle submatrices of the Hankel matrix An are nonsingular. As a useful application, then the rj’s yield the parameters from the three – term recurrence of the underlying polynomials. Via (5.5) the three – term recurrence can also be transferred to the case that calculations are carried out over finite fields.

So, let us assume from now on that all principal submatrices Ai, i ≤ n of the Hankel matrix An are nonsingular. For this case, Imamura and Yoshida [43] demonstrated that ` ` j j ` j − ` j+1 j δ j j = j−1 = 2 for even and j = j−1 = 2 for odd such that j is 1 if is odd j q2j (x) F x and 0 if is even ( u2j (x) then are the convergents to ( )). This means that there are only two possible recursions for uj(x) depending on the parity of j,namely

r2j r2j−1 2 u2j(x)=u2j−1(x) − xu2j−2(x),u2j−1(x)=u2j−2(x) − x u2j−4(x). r2j−1 r2j−3 So the algorithm is simpliﬁed in (5.7) and we obtain the recursion ! r2j u x − x −r2j−1x u x 2j( ) 1 r2j−1 2j−2( ) = 1 · . (5.10) v2j x x v2j−2 x ( ) r2j−1 0 ( )

By the above considerations we have the following three–term recurrence for u2j(x)(and also for q2j(x) with diﬀerent initial values).

r2j r2j−1 2 u2j(x)=(1− x)u2j−2(x) − x u2j−4(x). r2j−1 r2j−3 Since the Berlekamp - Massey algorithm determines the solution of equation (1.9) it must be

xj · u 1 t x . 2j(x)= j( ) F 1 as under (1.8). This is consistent with (1.16) where we consider the function ( x )rather than F (x). By the previous considerations, for tj(x), we have the recurrence

r2j r2j−1 tj(x)=(x − )tj−1(x) − tj−2(x)(5.11) r2j−1 r2j−3

the electronic journal of combinatorics 8 2001, #A1 25 Equation (5.11) now allows us to give a simple interpretation of the calculations in the single steps carried out in the course of the Berlekamp – Massey algorithm for the special case that all principle submatrices of the Hankel matrix An are nonsingular.

Proposition 5.1:LetAn be a Hankel matrix with real entries such that all principal submatrices Ai, i =1,...,n are nonsingular and let T be the linear operator mapping l T (x )=cl as in (1.9). Then for the parameters rj obtained via (5.6) it is

j−1 r2j−1 = T (x · tj−1(x)) = c0β1β2 ···βj−1 ,

j−1 r2j = αjT (x · tj−1(x)) = c0β1β2 ···βj−1αj, (5.12) where αj and β1,...,βj−1 are the parameters from the three-term recurrence of the orthogonal polynomials ti(x), i =0,...,j. Proof: The proposition, of course, follows directly from (5.11), since the three – term recurrence immediately yields the formula for the rj’s. Let us also verify the identities directly. From the considerations under (5.6) to (5.11) it is clear that the degree of u2j−2 is j − 1. Hence in this case b2j−2,j = b2j−2,j+1 = ...= b2j−2,2j−2 = 0 in (5.6) and

Xj−1 Xj−1 2j−2−t r2j−1 = b2j−2,tc2j−2−t = b2j−2,tT (x ) t=0 t=0 ! ! ! Xj−1 Xj−1 Xj−1 2j−2−t j−1 j−1−t j−1 t = T b2j−2,tx = T x b2j−2,tx = T x b2j−2,j−1−tx t=0 t=0 t=0 ! Xj−1 j−1 t j−1 = T x aj−1,tx = T (x tj−1(x)) = c0β1β2 ···βj−1 t=0 where the last equation follows by (5.3). A similar calculation shows that j r2j−1 j−1 j j−1 r2j = T x tj−1(x) − x tj−2(x) = T x tj−1(x) − βj−1x tj−2(x) r2j−3

r2j−1 βj−1 since by the previous calculation r2j−3 = . So by (5.3) further

r2j = c0β1β2 ···βj−1 [(α1 + α2 + ...+ αj) − (α1 + α2 + ...+ αj−1)] = c0β1β2 ···βj−1αj. £ Remarks: 1) Observe that with Proposition 5.1, the Berlekamp – Massey algorithm can be applied to determine the coeﬃcients αj and βj from the three – term recurrence of the orthogonal polynomials tj(x). From the parameters r2j−1 obtained by (5.6) in the odd steps of r2j−1 βj−1 αj the iteration = r2j−3 can be immediately calculated, and in the even steps = the electronic journal of combinatorics 8 2001, #A1 26 r2j β r2j−1 det(Aj )det(Aj−2) r is obtained. By (1.15) and (1.20) it is j−1 = r = 2 . Hence 2j−1 2j−3 det(Aj−1) det(Aj ) r2j−1 = , which means that the Berlekamp – Massey algorithm also yields a fast det(Aj−1) procedure to compute the determinant of a Hankel matrix. 2) By Proposition 5.1 the identity (5.6) reduces to Xj aj,tcj+t = c0β1β2 ···βj t=0

where the aj,t are the coeﬃcients of the polynomial tj(x), the βi’s are the coeﬃcients in their three – term recurrence and the ci’s are the corresponding moments. For the classical orthogonal polynomials all these parameters are usually known, such that one might also use (5.6) in the Berlekamp – Massey algorithm to derive combinatorial identities.

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